Mason's Gain Formula (With Examples) - Control Systems - Electrical Engineering (EE) PDF Download

The relation between an input variable and an output variable of a signal flow graph is given by Mason's Gain Formula.

For determination of the overall system transfer function, the gain is given by:

Where,

• P_k = gain of the k^th forward path.
• Δ = determinant of the graph, defined as 1 minus the sum of the gains of all individual loops plus the sum of the products of gains of all possible two non-touching loops minus the sum of the products of gains of all possible three non-touching loops, and so on. In compact form:

Δ = 1 - (sum of individual loop gains) + (sum of products of gains of two non-touching loops) - (sum of products of gains of three non-touching loops) + ...

Δ_k = value of Δ computed for the portion of the graph that does not touch the k^th forward path (i.e., exclude loops that touch that forward path when forming Δ_k).

Elements required to apply Mason's Gain Formula

Forward path

A forward path is any path from the input node to the output node in which no node is encountered more than once.

From the example signal flow graph shown below there are two forward paths with their path gains as indicated:

Loop

A loop is a closed path that starts and ends at the same node without passing through any other node more than once. The gain of a loop is the product of gains along that closed path.

There are five individual loops in the example signal flow graph with their loop gains as shown:

Non-touching loops

Two loops are said to be non-touching if they do not have any node in common. For Δ, products of gains of combinations of non-touching loops are used.

In the example graph there are two possible combinations of non-touching loops; their loop-gain products are shown here:

In the above SFG there are no combinations of three non-touching loops, four non-touching loops, and so on.

Example

Draw the Signal Flow Diagram and determine C/R for the block diagram shown in the figure.

The signal flow graph of the above diagram is drawn below

Sol.

P₁ = G₁G₂G₃  Δ₁ = 1

P₂ = -G₁G₄  Δ₂ = 1

Individual loops:

L₁ = - G₁G₂H₁

L₂ = - G₂G₃H₂

L₃ = - G₁G₂G₃

L₄ = G₁G₄

L₅ = G₄H₂

Non-touching loops = 0

Compute Δ for the whole graph.

Use the definition Δ = 1 - (sum of individual loop gains) + (sum of products of two non-touching loops) - ...

Since there are no non-touching loop combinations contributing, Δ = 1 - (L₁ + L₂ + L₃ + L₄ + L₅).

Substitute the loop gains:

Δ = 1 - [ -G₁G₂H₁ - G₂G₃H₂ - G₁G₂G₃ + G₁G₄ + G₄H₂ ]

Simplify the signs:

Δ = 1 + G₁G₂H₁ + G₂G₃H₂ + G₁G₂G₃ - G₁G₄ - G₄H₂

Apply Mason's Gain Formula:

Overall transfer C/R = (P₁Δ₁ + P₂Δ₂) / Δ

Substitute P₁, P₂, Δ₁ and Δ₂:

C/R = [ G₁G₂G₃ - G₁G₄ ] / [ 1 + G₁G₂H₁ + G₂G₃H₂ + G₁G₂G₃ - G₁G₄ - G₄H₂ ]

Factor if required for simplification:

C/R = G₁ [ G₂G₃ - G₄ ] / [ 1 + G₁G₂H₁ + G₂G₃H₂ + G₁G₂G₃ - G₁G₄ - G₄H₂ ]

Notes and stepwise procedure to apply Mason's Gain Formula

• Identify and list all forward paths from input to output and compute their path gains P_k.
• Identify all individual loops and compute each loop gain L_i.
• Determine all combinations of two non-touching loops and compute their product gains; repeat for three, four, ... non-touching loops as applicable.
• Compute Δ using the alternating sign series: 1 - (sum of individual loop gains) + (sum of products of gains of two non-touching loops) - (sum of products of gains of three non-touching loops) + ...
• For each forward path k, compute Δ_k by calculating Δ for the portion of the graph that excludes loops touching the k^th forward path.
• Compute overall transfer T = Σ (P_k Δ_k) / Δ.

Common remarks and applications

• Mason's Gain Formula yields an exact symbolic expression for the transfer function of a system represented by a signal flow graph and is especially useful for multi-loop feedback networks.
• Careful identification of non-touching loops is essential; missing or double-counting such combinations leads to incorrect Δ.
• When loops share nodes with a forward path, they are excluded from that path's Δ_k.
• The formula is widely used in control systems analysis, network reduction, and when deriving closed-loop transfer functions from block diagrams converted to signal flow graphs.